NAGOYA Univ.

NAGOYA UNIVERSITY

Aerospace Engineering

Control Systems Engineering

The stable manifold approach for optimal swing up and stabilization of an inverted pendulum with input

saturation

Ryu Fujimoto and Noboru Sakamoto

Nagoya University

IFAC World Congress 2011 August 31

NAGOYA Univ.

Outline

Background and problem formulation

Nonlinear optimal control

Approximation methods for HJ equations

Stable manifold approach

Simulation / Experiment results

Non-uniqueness of solution in HJ equations

Concluding remarks

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Inverted pendulum

Strong nonlinearity

Under actuated system

Benchmark problem of nonlinear controller design

Swing up and stabilization of inverted pendulum

Switching of swing up and stabilization controllers (Astrom & Furua ’00)

Topological issues / (Angeli ‘01, Astrom et.al ‘08)

Background/problem statement

This talk: Single optimal state feedback control Enhancement of valid approximation range for HJ eq.

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Inverted pendulum system

1

2222

11

222

21

2

41

2

4

1

2222

1

2

211

22

411

2

sin)(

cossinsin)()(

sin)(

sin)(cossincos

)(

xlmMmlmMJ

xxglmxxmlmlJxCmlJ

x

xlmMmlmMJ

xmglMmxxxlmxCxml

x

xf　　　　　　　　　　　

1

2222

2

2

1

2222

21

sin)(

)(

0

sin)(

cos0

)(

xlmMmlmMJ

CmlJ

xlmMmlmMJ

Cxml

xg 　　　　　　　

TT xxxxxxx ],,,[],,,[ 4321)()()( usatxgxfx

][18

][18

min

max

Vu

Vu

Input limitation:

0)(2

1sat)(

2

1sat 11

T

T

TT

T

x

VxgRR

x

VxgR

Qxxx

VxgRxgxf

x

V T

T

T

)(2

1sat)()( 1

Hamilton-Jacobi:

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・Taylor approximation method:

・Stable manifold approach：(N.Sakamoto and A.J.van der Schaft, IEEE TAC, 2008)

Approximate solutions for HJ eq

(○) is directly computed

(○) iterative method suitable for computer implementation

(○) Larger domain of convergence etc…

x

V

(×) Cannot handle non-analytic nonlinearities

(×) Computationally inefficient

(×) Small domain of convergence

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Stable manifold method for HJ eq.

NAGOYA Univ.

Stable manifold method for HJ eq.

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Closed loop trajectories for different iteration number

Computational result

-4 -3 -2 -1 0-1.5

-1

-0.5

0

0.5

1

1.5

pendulum angle(rad)

H

k=0(linear)k=6k=10k=25

Initial value

○

-4 -3 -2 -1 0-5

0

5

10

15

pendulum angle(rad)

an

gu

lar

velo

city

(ra

d/s

)

k=0

k=6

k=10

k=25

Hamiltonian Trajectories in the (θ,dθ/dt)-space

Initial value

○

(HJ eq.) QxxxpxgRxgxfxpH TTT

)()(

2

1sat)()()( 1

0)()(2

1sat)()(

2

1sat 11

xpxgRRxpxgR T

T

T

NAGOYA Univ. Simulation/Experiment results

Input voltage is under the limitation 18[V] Swing up with 2 swings → Robustness for parameter variations 20% →

Movie1

Movie2

Responses (simulation)

17.7[V]

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What do saturation functions do?

Effect of saturation function

Responses (simulation)

Saturation is an identity function inside of limitation

The solution solves the original HJ eq as well??

Qxxx

VxgRxgxf

x

V T

T

T

)(2

1sat)()( 1

0)(2

1sat)(

2

1sat 11

T

T

TT

T

x

VxgRR

x

VxgR

Hamilton-Jacobi eq with saturations

Simulation by HJ eq without saturation function

- Swing up control with 1 swing - Input voltage is far beyond 18[V]

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Analysis for a 2-dimensional model

Uniqueness of solution

Closed loop trajectories

Two layers of stable manifolds

Computed stable manifold without saturations

NAGOYA Univ. Uniqueness of solution

Closed loop trajectories

Computed stable manifold with saturations

One stable manifold is filtered out

Analysis for a 2-dimensional model

NAGOYA Univ. With smaller input constraint…

Swing up with 3 swings

Efficient strategy with low voltage

Third stable manifold → infinite layers

Movie3

Responses (simulation)

Apply 40 iterations of the stable

manifold algorithm

][12

][12

min

max

Vu

Vu

][18

][18

min

max

Vu

Vu

　

Feedback function is expressed

with polynomials

11.8[V]

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Concluding remarks

Optimal swing up and stabilization of inverted pendulum

Single feedback by solving a Hamilton-Jacobi equation

Large domain of validity to include the pending position

Explicitly include saturation functions in the HJ equation

1 swing, 2 swing and 3 swing controllers by changing the value of input limitation

An example of non-unique solutions to HJ equation

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Thank you for your attention

NAGOYA Univ. Stable manifold algorithm

N.Sakamoto and A.J.van der Schaft, 2008

),(),,( tptx kk converge to a solution on k),,0[

),(

),(

pxpFp

pxxFx

T

,2,1,0,,

,2,1,0,,

0,,

)(

1

01

00

kdsspsxetp

kdsspsxeetx

tpetx

tkk

stF

k

t

kk

stFFt

k

At

T

　　

　　

　　　

・・・(※)

A Hamiltoinan system

Stable F，smooth nonlinearities pxpx ,,,

NAGOYA Univ. Limitation of the Taylor method

,3 uxxx

1-dimensional optimal control problem

1)1(

02

1

2

1)(

223

223

xxxxp

xpxxpH

0

22 dtuxJ

Series solutions cannot approximate for

19.1x

has complex zeros at 19.1|| z1)1( 22 z

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Inverted pendulum setup